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whereN(¹
i;1) denotes the normal distribution with
mean¹
iand unit variance The signal¹=
(¹
1; : : : ; ¹
n) is sparse if most of the components¹
iare
zero Identifying the locations of the nonzero compo
nents based on the dataX= (X
1; : : : ; X
n) whennis
very large is a fundamental problem arising in many
applications, including fMRI (Genovese et al, 2002),
microarray analysis (Pawitan et al, 2005), and astro
nomical surveying (Hopkins et al, 2002) A common
approach in these problems entails coordinatewise
thresholding of the observed dataXat a given level,
identifying the locations whose corresponding obser
vation exceeds the threshold as signal components
Suppose that the number of nonzero components of¹
grows sublinearly innaccording ton1¡¯
for¯2(0;1),
and that each nonzero component takes the same
(positive) valuep
Suppose that instead of a single observation of a sparse
signal in noise, one were able to take multiple `looks,'
possibly adjusting the focus in a sequential fashion
Similar adaptive methods have been proposed in the
signal processing literature (Rangara jan, 2007), and
they certainly are conceivable in applications such as
{z
}
repeatedmtimes;
The paper is organized as follows In Section 2 we re
view the conventional nonadaptive approach to sparse
recovery We introduce our adaptive sensing tech
nique (DS) in Section 3, and in Section 4 we state our
main results, that DS enables the recoverability of sig
ni¯cantly weaker signals than standard, nonadaptive
methods Section 5 provides numerical simulations of
DS, and a short discussion appears in Section 6 Proofs
of the main results are given in the Appendix
2 SPARSE RECOVERY BY
NONADAPTIVE SENSING
Consider sparse signals havingn1¡¯
signal components
each of amplitudep
to estimate the locations of the signal components It
follows from techniques used in (Abramovich et al,
2006; Benjamini and Hochberg, 1995; Donoho and Jin,
2006; Donoho and Jin, 2008; Jin, 2003) that ifr > ¯,
the procedure (2), with a threshold¿that may depend
onr,¯, andn, drives both the FDP and NDP to zero
with probability one asn! 1 Conversely, ifr < ¯,
then no such coordinatewise thresholding procedure
can drive the FDP and NDP to zero simultaneously
with probability tending to one asn! 1 In other
words, for the speci¯ed signal parametrization and ob
servation model, the (¯ ; r) parameter plane is parti
tioned into two disjoint regions In the regionr > ¯,
sparse signal components can be reliably located us
ing a coordinatewise thresholding procedure In the
complementary region wherer < ¯, no coordinate
wise thresholding procedure is reliable in the sense of
controlling both the FDP and NDP This establishes
a sharp boundary in the parameter space,r=¯, for
largesample consistent recovery of sparse signals
3 DISTILLED SENSING
fori= 1;2; : : : ; nandj= 1;2; : : : ; k, where eachÁ(j)
i
is nonnegative, andZ(j)
iiid
» N(0;1) In addition, we
impose the restrictionP
i;jÁ(j)
i·n, limiting the to
tal amount of sensing energy Note that the standard
observation model (1) takes the form (3) withk= 1
andÁ(1)
i= 1 fori= 1; : : : ; n Another possibility is
to make multiple iid observations, but each with only
a fraction of the total sensing energy budget For ex
ample, setÁ(j)
i= 1=p
Number of observation stepsk;
Energy allocation strategy:E(j)
,P
k
j=1E(j)
·n;
forj= 1tokdo
X(j)
i=
(q
Output:
Final index setI(k)
;
Distilled observationsX(k)
DS:=fX(k)
i:i2I(k)
g;
Therefore, we are interested here in adaptive, sequen
tial designs offÁ(j)
ig
i;jthat tend to focus on the signal
components of¹ In other words, we allowÁ(j)
ito de
pend explicitly on the pastfÁ(`)
i; X(`)
ig
i;`